Let ρ: R→R be a continuous, strictly increasing function (so y>x ⇒ ρ(y) > ρ(x)). Show that d(x,y) = │ρ(y) – ρ(x)│ is a distance function on R, which is complete if and only if limx→∞ρ(x) = ∞ and limx→-∞ρ(x) = -∞.
here i have to show
- d is a distance function
- d is complete if and only if limx→∞ρ(x) = ∞ and limx→-∞ρ(x) = -∞.
is that right?
and what does it mean by 'distance function is complete'?